The Role of Diffraction in the Quantization of Dispersing Billiards

Details The Role of Diffraction in the Quantization of Dispersing Billiards The Role of Diffraction in the Quantization of Dispersing Billiards

1995-08-13

arxiv.org/abs/chao-dyn/9508005v1

Nonlinear Sciences

Chaotic Dynamics

12 pages, self-extracting uuencoded-tar-compressed PostScript file. In case of extraction problems, please write to Harel Prim

Scientific paper

10.1103/PhysRevLett.76.1615

We study diffraction corrections to the semiclassical spectral density of dispersing (Sinai) billiards. They modify the contributions of periodic orbits (PO's), with at least one segment which is almost tangent to the concave part of the boundary. Given a wavenumber $k$, all the PO's with length up to the Heisenberg length $O(k)$ are required for quantization. We show that most of the contributions of PO's which are longer than a limit $O(k^{2/3})$ must be corrected for diffraction effects. For orbits which just miss tangency, the corrections are of the same magnitude as the semiclassical contributions themselves. Orbits which bounce at extreme forward angles give very small terms in the standard semiclassical theory. The diffraction corrections increase their amplitude substantially.

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